Established physics · Quantum Mechanics 101 · Lesson 5 of 15
Observables: the question changes the basis
An observable identifies a measurable quantity; its measurement basis defines the alternatives used to calculate that experiment’s probabilities.
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Picture it
Send identically prepared, horizontally polarized photons into an ideal two-output polarization analyzer. Turn the analyzer. At 0°, every photon goes to the matching output. At 45°, the outputs share the events equally. You changed which alternatives the apparatus separates. You did not need to change the original preparation to change the question being asked.
An observable is a physical question
Position, momentum, energy, and a chosen spin component are different measurable quantities. In the idealized mathematical description, observables are represented by Hermitian operators. Their eigenvalues are the possible results, and their eigenstates supply the relevant alternatives. You can understand the apparatus idea before learning how to manipulate the matrices.
Basis change and physical change are different
Rewriting a state in a new basis changes its description. Rotating an analyzer changes the measurement performed. Physically rotating the prepared system is a third operation. These can be related mathematically, but they are different laboratory actions. Clear reasoning names which operation occurred, instead of saying vaguely that “the quantum state changed” whenever a probability changes.
Certainty belongs to the chosen measurement
An eigenstate gives a definite result for its associated ideal observable. That does not make every other observable definite. A horizontally polarized photon is definite for horizontal/vertical analysis, but has two possible outcomes for diagonal analysis. The same preparation therefore supports certainty about one question and a probability distribution for another.
Averages need not be possible single results
An expectation value is the probability-weighted average of outcomes over repeated preparations. If a measurement has only results +1 and −1, an expectation value of zero does not mean a single run returns zero. It means the balance of many results averages to zero. This distinction matters when reading both quantum predictions and experimental summaries.
- Keep the preparation fixed
- Choose the analyzer basis
- Calculate its outcome probabilities
A new measurement question can produce a new table for the same prepared state.
Try the example on paper
For horizontal polarization: analyzer angles 0°, 30°, 45°, and 90° give aligned-output probabilities 1, 3/4, 1/2, and 0. The orthogonal output carries the complementary probability.
Make it concrete
For a horizontally polarized photon and analyzer angle 30°, cos²30° = 3/4, giving 75% at the aligned output and 25% at its orthogonal partner. At 45° the probabilities are 50/50. An ideal analyzer separates two output channels; this demonstration is not counting photons absorbed by an ordinary single-output sheet polarizer.
The precise version
For an ideal observable A with eigenstates |aᵢ⟩, P(aᵢ) = |⟨aᵢ|ψ⟩|² and ⟨A⟩ = Σᵢ aᵢP(aᵢ). A linear-polarization state at angle 0° measured in an orthogonal basis rotated by θ has probabilities cos²θ and sin²θ. These are polarization angles; the spin-½ direction formula in the next lesson uses a half-angle instead.
Watch the trap
Do not confuse a coordinate relabeling with a physical measurement or rotation. Also, an average result between two allowed outcomes is not necessarily an allowed outcome itself.
Keep this
Always state the observable and basis. The same preparation can give different probability tables for different measurements.
Words for this lesson: observable · basis · eigenstate · expectation value · measurement · Born rule
Check your understanding
Reason through each question before opening its answer and explanation.
1. What defines which alternatives a chosen ideal measurement separates?
- The observer’s preferred interpretation
- The number of past trials
- Its measurement basis
Show answer and explanation
Its measurement basis
The physical observable and its basis determine the outcome alternatives for the probability calculation.
2. A state is an energy eigenstate. What is guaranteed for an ideal energy measurement?
- No time evolution of any amplitude phase
- A definite energy eigenvalue
- A definite value of every other observable
Show answer and explanation
A definite energy eigenvalue
An eigenstate supplies a definite value for that observable, without assigning definite values to all others.
3. A measurement has only +1 and −1 outcomes, each equally likely. What is its expectation value?
- 0, although no single outcome is 0
- 0, so every event returns 0
- 2
Show answer and explanation
0, although no single outcome is 0
The weighted average is (+1)/2 + (−1)/2 = 0. The allowed individual outcomes remain +1 and −1.
4. For horizontal polarization and a 45° ideal two-output analyzer, what are the probabilities?
- 50% and 50%
- 25% and 25%
- 100% and 0%
Show answer and explanation
50% and 50%
Cos²45° and sin²45° both equal 1/2.
5. Does expressing the same state in new coordinates necessarily perform a physical measurement?
- Only for position coordinates
- Yes, every rewritten amplitude is a detector record
- No; representation and laboratory interaction are different
Show answer and explanation
No; representation and laboratory interaction are different
A mathematical basis change can describe the same state without any physical apparatus interaction.
6. Why should the polarization angle formula not be copied blindly into the next spin lesson?
- Different physical examples relate angles to amplitudes differently
- Probability rules stop applying to spin
- Spin probabilities can exceed one
Show answer and explanation
Different physical examples relate angles to amplitudes differently
Linear-polarization analysis uses cos²θ; spin-½ along spatially separated axes uses cos²(θ/2). Both use squared amplitudes.